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Cognitive Development of Proof

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Proof and Proving in Mathematics Education

Part of the book series: New ICMI Study Series ((NISS,volume 15))

Abstract

This chapter traces the long-term cognitive development of mathematical proof from the young child to the frontiers of research. It uses a framework building from perception and action, through proof by embodied actions and classifications, geometric proof and operational proof in arithmetic and algebra, to the formal set-theoretic definition and formal deduction. In each context, proof develops over the long-term from the recognition and description of observed properties and the links between them, the selection of specific properties that can be used as definitions from which other properties may be deduced, to the construction of ‘crystalline concepts’ whose properties are a consequence of the context. These include Platonic objects in geometry, symbols having relationships in arithmetic and algebra and formal axiomatic systems whose properties are determined by their definitions.

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Author information

Authors and Affiliations

  1. Mathematics Education Research Centre, University of Warwick, Coventry, UK

    David Tall

  2. Department of Mathematics & Computing, University of Southern Queensland, Toowoomba, Australia

    Oleksiy Yevdokimov

  3. Department of Education in Technology and Science, Technion – Israel Institute of Technology, Haifa, Israel

    Boris Koichu

  4. Department of Mathematics and Statistics, York University, Toronto, Canada

    Walter Whiteley

  5. Faculty of Education and Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, Canada

    Margo Kondratieva

  6. Department of Mathematics, Taipei Municipal University of Education, Taipei, Taiwan

    Ying-Hao Cheng

Authors
  1. David Tall
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  2. Oleksiy Yevdokimov
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  3. Boris Koichu
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  4. Walter Whiteley
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  5. Margo Kondratieva
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  6. Ying-Hao Cheng
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Corresponding author

Correspondence to David Tall .

Editor information

Editors and Affiliations

  1. Ontario Institute for Studies in Education (OISE), University of Toronto, Toronto, ON, Canada

    Gila Hanna

  2. Faculty of Education, University of KwaZulu-Natal, Edgewood Campus, Ashwood, South Africa

    Michael de Villiers

Additional information

NB: References marked with * are in F. L. Lin, F. J. Hsieh, G. Hanna, & M. de Villiers (Eds.) (2009). ICMI Study 19: Proof and proving in mathematics education. Taipei, Taiwan: The Department of Mathematics, National Taiwan Normal University.

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Tall, D., Yevdokimov, O., Koichu, B., Whiteley, W., Kondratieva, M., Cheng, YH. (2012). Cognitive Development of Proof. In: Hanna, G., de Villiers, M. (eds) Proof and Proving in Mathematics Education. New ICMI Study Series, vol 15. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-2129-6_2

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